Purcell's Swimmer

We make a self-propelling Purcell Swimmer and dip it in a highly viscous fluid.

Purcell's Description

Purcell imagines a swimmer made of three 'links' or rods connected by two joints. The geometry of the swimmer can be described at any given point in time by the two angles $\theta_1$ and $\theta_2$. It undergoes a cycle as shown below from $S_1$ to $S_5$ moving its arms alternately.

Geometry

We need to define the position of the rods and the motors. Let's construct Purcell's and place the end points of the rod in the $S_1$ configuration. The rod is parametrised by $\eta$ — the ratio of the body length to the arm length — and $\phi$ = $\max_t(\theta_1(t)) = \max_t(\theta_2(t))$.

Dynamics

Define the time vector.

Let's define $\tau_1 = [0,0,q_1(t)]$ and $\tau_2 = [0,0,q_2(t)]$.

In Purcell's paper, the swimmer moves one arm at a given time. This can be achieved by a constant torque difference between the two motors. To describe this motion, we need a parameter $P = |q_2(0)|$ which is the maximum torque exerted by one of the motors. For simplicity, we assume that the other motor exerts no torque when the first is switched on.

It Swims!

Purcell Strokes

We can try finding the torque difference required for a Purcell stroke by trial and error, but it would be easier to sample different $P$ values and see which corresponds to a Purcell stroke.

Defining $\theta_1$ and $\theta_2$ by Purcell's convention as in the figure above, we can come up with a function that must be equal to zero for a Purcell stroke after half a cycle:

$$\Delta \phi = (\phi-\theta_2)+(\phi+\theta_1)$$

Now, we can try sampling over the parameter space to see which values of $P$ give $\Delta\phi_{1/2} = 0$ where $\Delta \phi _{1/2}$ is $\Delta \phi$ after half a cycle is completed.

The nice one-to-one mapping simplifies our job by quite a bit. We can use bisection method to find $P_0$.